I'm not at my laptop (where BaseCamp resides) but have you tried selecting the Route, so that its text content opens, then select each line/waypoint in turn (so that the waypoints are circled on the map view window) and simply use Delete Key to remove that item.
Or, if that doesn't work, this will:
create a track from the route;
select that track listing;
delete the track points you don't need;
save what's left as a new track.
If you need that as a Route, then you can re-define a new set of waypoints over that new track.